4.9E: Exercises

Exercise $\PageIndex{1}$

True or False? If true, prove it. If false, find the true answer.

1. The doubling time for $\displaystyle y=e^{ct}$ is $\displaystyle (ln(2))/(ln(c))$.

2. If you invest $\displaystyle $500$, an annual rate of interest of $\displaystyle 3%$ yields more money in the first year than a $\displaystyle 2.5%$ continuous rate of interest.

Answer

True

3. If you leave a $\displaystyle 100°C$ pot of tea at room temperature $\displaystyle (25°C)$ and an identical pot in the refrigerator $\displaystyle (5°C)$, with $\displaystyle k=0.02$, the tea in the refrigerator reaches a drinkable temperature $\displaystyle (70°C)$ more than $\displaystyle 5$ minutes before the tea at room temperature.

4. If given a half-life of t years, the constant $\displaystyle k$ for $\displaystyle y=e^{kt}$ is calculated by $\displaystyle k=ln(1/2)/t$.

Answer

False; $\displaystyle k=\frac{ln(2)}{t}$

Exercise $\PageIndex{2}$

For the following exercises, use $\displaystyle y=y_0e^{kt}.$

1. If a culture of bacteria doubles in $\displaystyle 3$ hours, how many hours does it take to multiply by $\displaystyle 10$?

2. If bacteria increase by a factor of $\displaystyle 10$ in $\displaystyle 10$ hours, how many hours does it take to increase by $\displaystyle 100$?

Answer

$\displaystyle 20$ hours

3. How old is a skull that contains one-fifth as much radiocarbon as a modern skull? Note that the half-life of radiocarbon is $\displaystyle 5730$ years.

4. If a relic contains $\displaystyle 90%$ as much radiocarbon as new material, can it have come from the time of Christ (approximately $\displaystyle 2000$ years ago)? Note that the half-life of radiocarbon is $\displaystyle 5730$ years.

Answer

No. The relic is approximately $\displaystyle 871$ years old.

5. The population of Cairo grew from $\displaystyle 5$ million to $\displaystyle 10$ million in $\displaystyle 20$ years. Use an exponential model to find when the population was $\displaystyle 8$ million.

6. The populations of New York and Los Angeles are growing at $\displaystyle 1%$ and $\displaystyle 1.4%$ a year, respectively. Starting from $\displaystyle 8$ million (New York) and $\displaystyle 6$ million (Los Angeles), when are the populations equal?

Answer

$\displaystyle 71.92$ years

7. Suppose the value of $\displaystyle $1$ in Japanese yen decreases at $\displaystyle 2%$ per year. Starting from $\displaystyle $1=¥250$, when will $\displaystyle $1=¥1$?

8. The effect of advertising decays exponentially. If $\displaystyle 40%$ of the population remembers a new product after $\displaystyle 3$ days, how long will $\displaystyle 20%$remember it?

Answer

$\displaystyle 5$ days $\displaystyle 6$ hours $\displaystyle 27$minutes

9. If $\displaystyle y=1000$ at $\displaystyle t=3$ and $\displaystyle y=3000$ at $\displaystyle t=4$, what was $\displaystyle y_0$ at $\displaystyle t=0$?

10. If $\displaystyle y=100$ at $\displaystyle t=4$ and $\displaystyle y=10$ at $\displaystyle t=8$, when does $\displaystyle y=1$?

Answer

$\displaystyle 12$

11. If a bank offers annual interest of $\displaystyle 7.5%$ or continuous interest of $\displaystyle 7.25%,$ which has a better annual yield?

12. What continuous interest rate has the same yield as an annual rate of $\displaystyle 9%$?

Answer

$\displaystyle 8.618%$

13. If you deposit $\displaystyle $5000$at $\displaystyle 8%$ annual interest, how many years can you withdraw $\displaystyle $500$ (starting after the first year) without running out of money?

14. You are trying to save $\displaystyle $50,000$ in $\displaystyle 20$ years for college tuition for your child. If interest is a continuous $\displaystyle 10%,$ how much do you need to invest initially?

Answer

$6766.76

15. You are cooling a turkey that was taken out of the oven with an internal temperature of $\displaystyle 165°F$. After $\displaystyle 10$ minutes of resting the turkey in a $\displaystyle 70°F$ apartment, the temperature has reached $\displaystyle 155°F$. What is the temperature of the turkey $\displaystyle 20$ minutes after taking it out of the oven?

16. You are trying to thaw some vegetables that are at a temperature of $\displaystyle 1°F$. To thaw vegetables safely, you must put them in the refrigerator, which has an ambient temperature of $\displaystyle 44°F$. You check on your vegetables $\displaystyle 2$ hours after putting them in the refrigerator to find that they are now $\displaystyle 12°F$. Plot the resulting temperature curve and use it to determine when the vegetables reach $\displaystyle 33°$.

Answer

$\displaystyle 9$hours $\displaystyle 13$minutes

17. You are an archaeologist and are given a bone that is claimed to be from a Tyrannosaurus Rex. You know these dinosaurs lived during the Cretaceous Era ($\displaystyle 146$ million years to $\displaystyle 65$ million years ago), and you find by radiocarbon dating that there is $\displaystyle 0.000001%$ the amount of radiocarbon. Is this bone from the Cretaceous?

18. The spent fuel of a nuclear reactor contains plutonium-239, which has a half-life of $\displaystyle 24,000$years. If $\displaystyle 1$ barrel containing $\displaystyle 10kg$ of plutonium-239 is sealed, how many years must pass until only $\displaystyle 10g$ of plutonium-239 is left?

Answer

$\displaystyle 239,179$ years

Exercise $\PageIndex{3}$

For the next set of exercises, use the following table, which features the world population by decade.

Years since 1950	Population (millions)
0	2,556
10	3,039
20	3,706
30	4,453
40	5,279
50	6,083
60	6,849

Source: http://www.factmonster.com/ipka/A0762181.html.

1.The best-fit exponential curve to the data of the form $\displaystyle P(t)=ae^{bt}$ is given by $\displaystyle P(t)=2686e^{0.01604t}$. Use a graphing calculator to graph the data and the exponential curve together.

2. Find and graph the derivative $\displaystyle y′$of your equation. Where is it increasing and what is the meaning of this increase?

Answer

$\displaystyle P'(t)=43e^{0.01604t}$. The population is always increasing.

3. Find and graph the second derivative of your equation. Where is it increasing and what is the meaning of this increase?

4. Find the predicted date when the population reaches $\displaystyle 10$ billion. Using your previous answers about the first and second derivatives, explain why exponential growth is unsuccessful in predicting the future.

Answer

The population reaches $\displaystyle 10$ billion people in $\displaystyle 2027$.

Exercise $\PageIndex{4}$

For the next set of exercises, use the following table, which shows the population of San Francisco during the 19th century.

Years since 1850	Population (thousands)
0	21.00
10	56.80
20	149.5
30	234.0

Source: http://www.sfgenealogy.com/sf/history/hgpop.htm.

1. The best-fit exponential curve to the data of the form $\displaystyle P(t)=ae^{bt}$ is given by $\displaystyle P(t)=35.26e^{0.06407t}$. Use a graphing calculator to graph the data and the exponential curve together.

2. Find and graph the derivative $\displaystyle y′$ of your equation. Where is it increasing? What is the meaning of this increase? Is there a value where the increase is maximal?

Answer

$\displaystyle P'(t)=2.259e^{0.06407t}$. The population is always increasing.

3. Find and graph the second derivative of your equation. Where is it increasing? What is the meaning of this increase?